Average Error: 9.6 → 0.3
Time: 7.7s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1, -\mathsf{fma}\left(1, z \cdot y, \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1, -\mathsf{fma}\left(1, z \cdot y, \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r501754 = x;
        double r501755 = y;
        double r501756 = log(r501755);
        double r501757 = r501754 * r501756;
        double r501758 = z;
        double r501759 = 1.0;
        double r501760 = r501759 - r501755;
        double r501761 = log(r501760);
        double r501762 = r501758 * r501761;
        double r501763 = r501757 + r501762;
        double r501764 = t;
        double r501765 = r501763 - r501764;
        return r501765;
}
↓
double f(double x, double y, double z, double t) {
        double r501766 = y;
        double r501767 = log(r501766);
        double r501768 = x;
        double r501769 = z;
        double r501770 = 1.0;
        double r501771 = log(r501770);
        double r501772 = r501769 * r501766;
        double r501773 = 0.5;
        double r501774 = 2.0;
        double r501775 = pow(r501766, r501774);
        double r501776 = r501769 * r501775;
        double r501777 = pow(r501770, r501774);
        double r501778 = r501776 / r501777;
        double r501779 = r501773 * r501778;
        double r501780 = fma(r501770, r501772, r501779);
        double r501781 = -r501780;
        double r501782 = fma(r501769, r501771, r501781);
        double r501783 = t;
        double r501784 = r501782 - r501783;
        double r501785 = fma(r501767, r501768, r501784);
        return r501785;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.3
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.6

    \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

  2. Simplified9.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, z \cdot \log \left(1 - y\right) - t\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(z \cdot \log 1 - \left(1 \cdot \left(z \cdot y\right) + \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)} - t\right)\]

  4. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\mathsf{fma}\left(z, \log 1, -\mathsf{fma}\left(1, z \cdot y, \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right)} - t\right)\]

  5. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(z, \log 1, -\mathsf{fma}\left(1, z \cdot y, \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))